Information for Graduate Students
I am looking for graduate students interested to do research in analysis and PDE.
First, what do I do? My work is in the area that involves Partial Differential Equations, Fourier Analysis and Functional Analysis.
Occasionally I also need to run a numerical simulation or two, mainly to gain intuition for proofs
(I am not very good at numerics though). I prefer problems that originate in physics or other sciences.
I am working on problems motivated by Fluid Mechanics, Quantum Mechanics, Combustion, Mathematical Biology and even Finance.
I can't say I am fully applied mathematician - I work not as much on concrete and detailed problems that would have direct links
to technology, but more on conceptual models of important processes. This way, you can have both beautiful, deep and rich mathematics
and the feeling that you understand something new about the real world. This mix is the most delicious I have ever tasted!
I have been moving into more applied directions recently though.
PDE is a very active and vibrant field. It is certainly one of the
central themes of modern mathematics, with connections to numerous
other branches of mathematics. PDEs are often motivated by
applications, and if you are working on PDEs you will likely find
yourself interacting with scientists from other disciplines.
Often, there is a certain learning curve one has to master to work
on PDEs. At first, the area may seem a bit technical and too
varied, even messy. However, after covering this curve the subject
becomes very rewarding. Once central ideas and connections become
clear, technicality and variety turn into depth and richness.
Learning some new partial differential equation well becomes like
making a new good friend - and the fact that there are many
different PDEs becomes a big positive. Another positive aspect of working in
PDE is that PDE experts are in high demand in academia. Should you
decide to transition to industry, PDE background will also be helpful due
to connections to applications.
If you are interested in analysis and PDE or undecided on what research you would like to do, please feel free
to email me or stop by so that we can chat. In order that this conversation can have more substance, let me state
some principles which I follow when mentoring research of a junior colleague. I will describe some of the directions
of my research in more detail after that.
- I think it important that you have
some research problems in mind early. Even if you need to learn
and read through some papers and books before starting research,
having a problem in mind can significantly increase your
- While it is important to get into research mode early, it is
also important to accumulate enough knowledge to start seeing
various directions and open problems around your work. This helps
you find best directions to work on and build more relationships
and collaborations with other mathematicians later.
- Career planning is important. It is best to learn about all
tracks available to you: research, teaching or industry. It is
good to have some idea about your most likely direction at least a
year or two before graduation.
- Everybody is different. What works for me may not work for
you and other way around.
When I was in graduate school, abstract algebra was hard for me. It is difficult for me to prove something
if I can't draw a picture or build a simplified model example to work out first. Fortunately, midway through the first year,
I became friends with another student, who was great in algebra. He just somehow knew what to do. But he did not like geometry.
He could not fathom how someone can prove something by using pictures. Ever since, we collaborated. Life became wonderfully
easy, and we learned a lot from each other. Moral? Talk to others about your math
strengths and struggles. How I handle math may not work for you, but I likely have seen enough styles of doing math to give you some
- As should be clear from the previous paragraph, mathematics
research benefits greatly from talking to people about
mathematics. I usually meet with my students or postdocs at least
once a week to discuss math. We have one of the nation's strongest
groups in analysis and PDE at Madison by research reputation (US
News has ranked us 7th and 10th in the nation in most recent
rankings), so there are plenty of high quality people for you to
Now a bit more about my research. Partial Differential equations
is an exceedingly diverse field, with very different subareas. The
reason is that many processes in nature seem to be described well
by PDE - a realization that probably goes back to Newton. Since
nature is so rich, there are many different kinds of partial
differential equations, and different tools are needed to analyze
them. Calculus of variations, comparison principles for the
solutions, Fourier analysis, functional analysis, stochastic calculus, ideas from
geometry are just some examples of the tool sets that are used to
analyze PDEs. So as someone working in PDEs, I use whatever it
takes. There are two ideals that I usually try to keep in mind. I
like it when mathematics is rich and beautiful, and I like when it
refers to or is at least motivated by real life phenomena -
problems connected to physics, biology or economics. It is not
always easy to balance these two aspirations! One of my favorite
tools is Fourier analysis, because it is so abundant with deep and
elegant techniques and ideas.
After being pioneered by Joseph Fourier in early 1900s, Fourier analysis went on to become one of the dominant themes in mathematics and its applications for the next two centuries.
It is applied widely to partial differential equations (Fourier transform turns differentiation into multiplication, and that is often beneficial -
but applications get a lot richer and subtler than that). It often plays an important role in numerical simulations of PDEs, too. Fourier analysis
methods are also used heavily in number theory, ergodic theory and occasionally in almost every other area of mathematics. The methods continue to develop and
grow more sophisticated. Out of the four 2010 Fields medalists, for example, Smirnov is a Fourier analyst by training, and Lindenstrauss uses Fourier
analysis methods extensively in his work -
even though they got their awards for work done in other areas (percolation theory and statistical physics for Smirnov, ergodic theory and number theory for Lindestrauss).
The set of ideas and tools that you learn in Fourier analysis just seems to be extremely adaptable and useful in all sorts of problems. Learning these techniques
prepares your brain for virtually anything mathematical in a very efficient way.
In any case here are several areas that are currently central in my research.
Mathematical Fluid Mechanics
Mixing and enhancement of diffusion by fluid flow
Quantum mechanics, Schrodinger operators and spectral theory